In
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...
, the second fundamental form (or shape tensor) is a
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to ...
on the
tangent plane
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
of a
smooth surface
In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric.
Surfaces have been extensively studied from various perspective ...
in the three-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
, usually denoted by
(read "two"). Together with the
first fundamental form, it serves to define extrinsic invariants of the surface, its
principal curvature
In differential geometry, the two principal curvatures at a given point of a surface are the maximum and minimum values of the curvature as expressed by the eigenvalues of the shape operator at that point. They measure how the surface bends b ...
s. More generally, such a quadratic form is defined for a smooth immersed
submanifold
In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which ...
in a
Riemannian manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ...
.
Surface in R3
Motivation
The second fundamental form of a
parametric surface A parametric surface is a surface in the Euclidean space \R^3 which is defined by a parametric equation with two parameters Parametric representation is a very general way to specify a surface, as well as implicit representation. Surfaces that occ ...
in was introduced and studied by
Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
. First suppose that the surface is the graph of a twice
continuously differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
function, , and that the plane is
tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
to the surface at the origin. Then and its
partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Pa ...
s with respect to and vanish at (0,0). Therefore, the
Taylor expansion
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor seri ...
of ''f'' at (0,0) starts with quadratic terms:
:
and the second fundamental form at the origin in the coordinates is the
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to ...
:
For a smooth point on , one can choose the coordinate system so that the plane is tangent to at , and define the second fundamental form in the same way.
Classical notation
The second fundamental form of a general parametric surface is defined as follows. Let be a regular parametrization of a surface in , where is a smooth
vector-valued function of two variables. It is common to denote the partial derivatives of with respect to and by and . Regularity of the parametrization means that and are linearly independent for any in the domain of , and hence span the tangent plane to at each point. Equivalently, the
cross product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and ...
is a nonzero vector normal to the surface. The parametrization thus defines a field of unit normal vectors :
:
The second fundamental form is usually written as
:
its matrix in the basis of the tangent plane is
:
The coefficients at a given point in the parametric -plane are given by the projections of the second partial derivatives of at that point onto the normal line to and can be computed with the aid of the
dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alg ...
as follows:
:
For a
signed distance field of
Hessian , the second fundamental form coefficients can be computed as follows:
:
Physicist's notation
The second fundamental form of a general parametric surface is defined as follows.
Let be a regular parametrization of a surface in , where is a smooth
vector-valued function of two variables. It is common to denote the partial derivatives of with respect to by , . Regularity of the parametrization means that and are linearly independent for any in the domain of , and hence span the tangent plane to at each point. Equivalently, the
cross product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and ...
is a nonzero vector normal to the surface. The parametrization thus defines a field of unit normal vectors :
:
The second fundamental form is usually written as
:
The equation above uses the
Einstein summation convention
In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of ...
.
The coefficients at a given point in the parametric -plane are given by the projections of the second partial derivatives of at that point onto the normal line to and can be computed in terms of the normal vector as follows:
:
Hypersurface in a Riemannian manifold
In
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
, the second fundamental form is given by
:
where is the
Gauss map, and the
differential of regarded as a
vector-valued differential form, and the brackets denote the
metric tensor
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allow ...
of Euclidean space.
More generally, on a Riemannian manifold, the second fundamental form is an equivalent way to describe the
shape operator
In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric.
Surfaces have been extensively studied from various perspective ...
(denoted by ) of a hypersurface,
:
where denotes the
covariant derivative
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differe ...
of the ambient manifold and a field of normal vectors on the hypersurface. (If the
affine connection
In differential geometry, an affine connection is a geometric object on a smooth manifold which ''connects'' nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values i ...
is
torsion-free, then the second fundamental form is symmetric.)
The sign of the second fundamental form depends on the choice of direction of (which is called a co-orientation of the hypersurface - for surfaces in Euclidean space, this is equivalently given by a choice of
orientation of the surface).
Generalization to arbitrary codimension
The second fundamental form can be generalized to arbitrary
codimension
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties.
For affine and projective algebraic varieties, the codimension equals ...
. In that case it is a quadratic form on the tangent space with values in the
normal bundle and it can be defined by
:
where
denotes the orthogonal projection of
covariant derivative
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differe ...
onto the normal bundle.
In
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
, the
curvature tensor of a
submanifold
In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which ...
can be described by the following formula:
:
This is called the
Gauss equation, as it may be viewed as a generalization of Gauss's
Theorema Egregium.
For general Riemannian manifolds one has to add the curvature of ambient space; if is a manifold embedded in a
Riemannian manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ...
then the curvature tensor of with induced metric can be expressed using the second fundamental form and , the curvature tensor of :
:
See also
*
First fundamental form
*
Gaussian curvature
In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point:
K = \kappa_1 \kappa_2.
The Gaussian radius of curvature is the reciprocal of .
...
*
Gauss–Codazzi equations
*
Shape operator
In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric.
Surfaces have been extensively studied from various perspective ...
*
Third fundamental form
In differential geometry, the third fundamental form is a surface metric denoted by \mathrm. Unlike the second fundamental form, it is independent of the surface normal.
Definition
Let be the shape operator and be a smooth surface. Also, l ...
*
Tautological one-form
References
*
*
*
External links
* Steven Verpoort (2008
Geometry of the Second Fundamental Form: Curvature Properties and Variational Aspectsfrom
Katholieke Universiteit Leuven
KU Leuven (or Katholieke Universiteit Leuven) is a Catholic research university in the city of Leuven, Belgium. It conducts teaching, research, and services in computer science, engineering, natural sciences, theology, humanities, medicine, ...
.
{{curvature
Differential geometry
Differential geometry of surfaces
Riemannian geometry
Curvature (mathematics)
Tensors